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GARCH models are commonly used as latent processes in econometrics, Þnancial economics and macroeconomics. Yet no exact likelihood analysis of these models has been provided so far. In this paper we outline the issues and suggest a Markov chain Monte Carlo algorithm which allows the calculation of a classical estimator via the simulated EM algorithm or a Bayesian solution in O(T ) computational operations, where T denotes the sample size. We assess the performance of our proposed algorithm in the context of both artiÞcial examples and an empirical application to 26 UK sectorial stock returns, and compare it to existing approximate solutions.

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We develop generalized indirect estimation procedures that handle equality and inequality constraints on the auxiliary model parameters by extracting information from the relevant multipliers, and compare their asymptotic efficiency to maximum likelihood. We also show that, regardless of the validity of the restrictions, the asymptotic efficiency of such estimators can never decrease by explicitly considering the multipliers associated with additional equality constraints. Furthermore, we discuss the variety of effects on efficiency that can result from imposing constraints on a previously unrestricted model. As an example, we consider a stochastic volatility process estimated through a GARCH model with Gaussian or t distributed errors. 945

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